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Summary and Conclusions

We contribute to the systemic risk measurement literature by developing formal hypothesis test statistics that can be used to detect systemic risk using ∆CoVaR and MES estimates based on daily stock return data. Our test statistics are based on the null hypothesis of Gaussian stock returns, however our methodology can be generalized other distributions that exhibit asymptotic tail independence. We use Monte Carlo simulation to estimate the critical values of the sampling distributions of our proposed test statistics and use these critical values to test for evidence of systemic risk in a broad cross section of stocks using daily return data over the period 2006-2007.

While we are the first to introduce formal hypothesis tests for detecting systemic risk using

∆CoVaR and MES estimates, our hypothesis tests are composite tests, and they do not provide

27 A way to mitigate the power issue could be to employ intraday returns in the estimation of systemic risk proxies.

As a result, the number of return series used in the estimation will increase and the nonparametric estimators can become more efficient.. For instance, Zhang, Zhou, Zhu (2009) show that volatility measures based on high-frequency tick data can better explain the likelihood of tail events, such as default probabilities. A caveat of this approach is that trading noise in tick data might adversely bias the systemic risk estimates.

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an ideal solution to the testing problem. Our hypothesis tests are joint test because they require the adoption of a specific return distribution under the null hypothesis. The joint nature of the null hypothesis is problematic because there is a wide range of distributions that exhibit asymptotic tail independence. The joint nature of the test complicates the interpretation of test rejections. For example, we show that, depending on the return generating process, our tests may reject the null hypothesis of no systemic risk when returns are generated by skewed, but tail-independent distributions. Thus, the choice of a specific return distribution to characterize returns under the null hypothesis is a crucial aspect of systemic risk test design. While this finding suggests the need for additional research focused on clarifying the null hypothesis used to characterize stock return distributions, our findings suggest that ∆CoVaR and MES tests are likely to have only weak power, so further efforts to refine this test may provide only limited improvements in test performance.

In particular, our simulation results suggest that nonparametric ∆CoVaR and MES statistics are unlikely to detect asymptotic tail dependence unless the tail dependence is strong. And even then, the power of these test statistics is limited by large variation in sampling distribution of nonparametric estimators. If systemic risk is truly manifested as asymptotic left-tail dependence in stock returns, our analysis suggests that ∆CoVaR and MES measures may be incapable of reliably detecting a firm’s systemic risk potential.

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Appendix

A.1 Parametric Estimator for ∆CoVaR when Returns are Gaussian

In this section, we derive the CoVaR measure for the returns on a market portfolio, 𝑅̃𝑀, conditional on a specific stock return, 𝑅̃𝑗, equal to its 1 percent value at risk. The market portfolio conditional return distribution is given by,

𝑅̃𝑀| (𝑅̃𝑗 = G−1(. 01, 𝑅̃𝑗)) ~ 𝑁 [𝜇𝑀+ 𝜌𝜎𝜎𝑀

𝑗−1(. 01, 𝑅̃𝑗) − 𝜇𝑗), (1 − 𝜌𝑗𝑀2 )𝜎𝑀2] , (A1) where G−1(. 01, 𝑅̃𝑗) represents the inverse cumulative normal distribution of the unconditional return 𝑅̃𝑗 evaluated at the 0.01 cumulative probability. Using, G−1(. 01, 𝑅̃𝑗) = 𝜇𝑗− 2.32635 𝜎𝑗, the 1 percent CoVaR for the market conditional on 𝑅̃𝑗 equal to its 1 percent VaR is,

𝐶𝑜𝑉𝑎𝑅 (𝑅̃𝑀| (𝑅̃𝑗 = Φ−1(. 01, 𝑅̃𝑗))) = 𝜇𝑀− 𝜌𝑗𝑀𝜎𝜎𝑀

𝑗 (2.32635 𝜎𝑗) − 2.32635 𝜎𝑀√1 − 𝜌𝑗𝑀2 . (A2) Similarly, the return distribution for 𝑅̃𝑀, conditional on 𝑅̃𝑗 equal to its median is,

𝑅̃𝑀| (𝑅̃𝑗 = G−1(. 50, 𝑅̃𝑗)) ~ 𝑁[𝜇𝑀, (1 − 𝜌𝑗𝑀2 )𝜎𝑀2]. (A3) Consequently, the CoVaR for the portfolio with 𝑅̃𝑗 evaluated at its median return is,

𝐶𝑜𝑉𝑎𝑅 (𝑅̃𝑀| (𝑅̃𝑗 = G−1(. 50, 𝑅̃𝑗))) = 𝜇𝑀− 2.32635𝜎𝑀√1 − 𝜌𝑗𝑀2 , (A4)

Subtracting (A4) from (A2) and defining 𝛽𝑗𝑀 = 𝐶𝑜𝑣(𝑅̃𝜎𝑗,𝑅̃𝑀)

𝑀2 , the so-called contribution CoVaR measure, ∆CoVaR is28,

∆𝐶𝑜𝑉𝑎𝑅 (𝑅̃𝑀| (𝑅̃𝑗 = G−1(. 01, 𝑅̃𝑗))) = − 𝛽𝑗𝑀∙ 2.32635 𝜎𝜎𝑀2

𝑗 (A5a)

= −𝜌𝑗𝑀∙ 2.32635𝜎𝑀 (A5b)

28 Reversing the order of the conditioning variable (i.e., the CoVaR for 𝑅̃𝑗 conditional on 𝑅̃𝑀 equal to its 1 percent VaR), it is straight-forward to show that the so-called exposure CoVaR measure is,

∆𝐶𝑜𝑉𝑎𝑅 (𝑅̃𝑖| (𝑅̃𝑀= G−1(. 01, 𝑅̃𝑀))) = − 𝛽𝑗𝑀∙ 2.32635 𝜎𝑀

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A.2 Parametric MES Estimator for Gaussian Returns

The marginal expected shortfall measure is the expected shortfall calculated from a conditional return distribution. The conditioning event is the return on a market portfolio, 𝑅̃𝑀, less than or equal to its 5 percent VaR value.

Under the assumption of bivariate normality, the conditional stock return is normally distributed, and consequently,

𝐸(𝑅̃𝑗|𝑅̃𝑀 = 𝑟𝑀) = 𝜇𝑗− 𝜌𝜎𝜎𝑗

𝑀 𝜇𝑀+ 𝜌𝜎𝜎𝑗

𝑀𝑟𝑀. (A6) Now, if 𝑅̃𝑀 is normally distributed with mean 𝜇𝑀 and standard deviation 𝜎𝑀, then the expected value of the market return truncated above the value “b” is,

𝐸(𝑅̃𝑀|𝑅̃𝑀 < 𝑏) = 𝜇𝑀 − 𝜎𝑀[𝜙( where the constant (2.062839) is a consequence of the 5 percent tail conditioning on the market return, i.e., Φ(−1.645)𝜙(−1.645)= 2.062839.

A.3 Can 𝑏̂2(∆𝐶𝑜𝑉𝑎𝑅) − 𝑏̂1(∆𝐶𝑜𝑉𝑎𝑅) and 𝑏̂2(𝑀𝐸𝑆) − 𝑏̂1(𝑀𝐸𝑆) Detect Systemic Risk?

We use Monte Carlo simulation to confirm the intuition that the difference between the nonparametric and the parametric ΔCoVaR and MES estimators can detect systemic risk. In particular, under the alternative hypothesis that stock returns have asymptotic tail dependence, we expect the nonparametric ΔCoVaR and MES estimators to produce larger negative values relative to their parametric estimator counterparts.

Table A1 reports the results of a Monte Carlo exercise in which the sampling distribution of the difference in the nonparametric and parametric ∆CoVaR (and MES) estimators is constructed for a sample size of 500 for bivariate Gaussian and Student-t return generating processes. While the

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numerical estimates of the sampling distribution quantiles in Table A1 are specific to the population parameter values used in the simulation, the qualitative results generalize. When returns are generated by the Student-t distribution, a distribution with asymptotic tail

dependence, the nonparametric ∆CoVaR and MES estimators remain consistent and tend to produce larger negative values compared to their parametric counterparts. As a consequence, the quantiles of sampling distribution of the difference between the nonparametric and parametric estimators are shifted to the left under alternative hypothesis that include asymptotic tail dependence.

A.4 Small Sample Properties of the 𝐷𝑊𝐻𝑇∆𝐶𝑜𝑉𝑎𝑅 and 𝐷𝑊𝐻𝑇𝑀𝐸𝑆 Test Statistics

To better understand the small sample properties of the 𝐷𝑊𝐻𝑇∆𝐶𝑜𝑉𝑎𝑅 and 𝐷𝑊𝐻𝑇𝑀𝐸𝑆 test statistics, we performed a Monte Carlo simulation to estimate the distributions of these test statistics in a sample size of 500 daily stock returns.

Our Monte Carlo estimates were constructed as follows. For both ∆𝐶𝑜𝑉𝑎𝑅 and MES

simulations, we set the reference portfolios equal to the equally-weighted CRSP market return portfolio, 𝑅̃𝑃 = 𝑅̃𝑀. We draw a set of three bivariate Gaussian parameter values (𝜌𝑗𝑀, 𝜎𝑀, 𝜎𝑗)

Table A1: The Sampling Distribution of the Difference Between the Nonparameteric and Parametric ∆CoVaR and MES Estimators under Gaussian and Alternative

Student t Return Generating Process

Monte Carlo Estimates of the sample distributions of the difference between the nonparametric and the parametric ΔCoVaR and MES estimators under two alternative return generating provcess: a bivariate Gaussian (the null hypothesis) and a bivariate Student-t (an alternative hypothesis with asymptotic tail dependence). Both return generating process have zero mean, return standard deviations of 0.20 for both components, and returns correlation of 0.5. The Monte Carlo quantile estimates are based on a sample size of 500, with 2000 replications and the output smoothed using a Gaussian Kernel density estimator.

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from a set of three possible values that are characteristic of 10 percentile, median, and 90 percentile values of the parameters observed in the population of stock returns: 𝜌𝑗𝑀 =

{. 17, .42, .58}; 𝜎𝑀 = {. 014, .023, .037}; and 𝜎𝑗 = {. 005, .008, .03}. 29 On each Monte Carlo replication, we randomly select one parameter value from each set, where the probability of selecting any one of the three possible parameters is 1/3.

Once the Gaussian parameters are selected, a sample of 500 daily bivariate Gaussian returns is simulated and the sample parametric and nonparametric ∆𝐶𝑜𝑉𝑎𝑅 and MES estimators are

calculated. The 500 simulated returns are then resampled 500 times using the bootstrap technique to calculate sample estimates for 𝑉𝑎𝑟 (𝑏̂𝑗(∆𝐶𝑜𝑉𝑎𝑅)) and 𝑉𝑎𝑟 (𝑏̂𝑗(𝑀𝐸𝑆)), for 𝑗 = 1,2. The 𝐷𝑊𝐻𝑇∆𝐶𝑜𝑉𝑎𝑅 and 𝐷𝑊𝐻𝑇𝑀𝐸𝑆 test statistics are calculated. The process then repeats 25,000 times with parameter selection, simulation, estimation, and the bootstrap to generate a Monte Carlo sampling distribution for the 𝐷𝑊𝐻𝑇∆𝐶𝑜𝑉𝑎𝑅 and 𝐷𝑊𝐻𝑇𝑀𝐸𝑆 test statistics.

29 Percentile distribution for the market volatility is based on a historical distribution between 1990 and 2007.

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Industry Identifying Information Number of Firms Mean DCoVaR Mean MES

Broker Dealers SIC 62 55 -0.0321 -0.0102

Construction SIC 15,16,17 41 -0.0345 -0.0095

Depository Institutions SIC 60; NAICS 5221, 551111; FRBNY * 426 -0.0196 -0.0067

Insurance SIC 63,64 138 -0.0206 -0.0075

Manufacturing SIC 20-39 1336 -0.0231 -0.0076

Mining SIC 10-14 151 -0.0272 -0.0100

Other Financial SIC 61,65,67: NAICS 52 74 -0.0267 -0.0087

Public Administration SIC 91-99 11 -0.0085 -0.0033

Retail Trade SIC 52-59 227 -0.0231 -0.0075

Services SIC 70,72,73,75,76,78,79,80-89 626 -0.0213 -0.0070

Transportation, Communications SIC 40-49 320 -0.0211 -0.0086

Wholesale Trade SIC 50-51 113 -0.0234 -0.0084

Table 1: Sample Characteristics by Industrial Classification

∆CoVaR is estimated as the difference between two linear quantile regressions estimates of the market portfolio return on individual stock returns: the 1 percent quantile estimate less the 50 percent quantile estimate. The reference portfolio is the CRSP equally-weighted market index. Quantile regressions are estimated using the R package Quantreg. MES is estimated as the average of individual stock returns on sample subset of days that correspond with the 5 percent worst days of the equally-weighted CRSP stock market index. SIC is the standard industrical classification. NAICS is the North American Industry Classicification System.

*Depository institutions are identified using the Bank Holding Company (BHC) dataset prepared by the Federal Reserve Bank of New York (FRBNY, March 18, 2008). http://www.newyorkfed.org/research/banking_research/datasets.html. We supplement this definition adding institutions that either have a 2-digit SIC of “60”, or have a 4-digit NAICS of “5221” or a 6-digit NAICS or “551111” in the depository definition.

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15 CITIZENS FIRST BANCORP INC 203,931 Depository Institutions -0.0187

16 M S C INDUSTRIAL DIRECT INC 2,149,047 Manufacturing -0.0186

17 SMURFIT STONE CONTAINER CORP 2,928,624 Manufacturing -0.0184

18 SOUTHERN COPPER CORP 16,450,064 Mining -0.0180

19 STILLWATER MINING CO 1,095,439 Mining -0.0180

20 A M R CORP DEL 5,753,965 Transportation, Communications -0.0179

21 AMERIPRISE FINANCIAL INC 13,095,030 Broker Dealers -0.0178

22 NASDAQ STOCK MARKET INC 3,617,941 Broker Dealers -0.0178

23 U A L CORP 4,264,734 Transportation, Communications -0.0177

24 PARKER HANNIFIN CORP 9,950,826 Manufacturing -0.0176

33 EMPIRE RESOURCES INC DEL 104,041 Wholesale Trade -0.0173

34 CLEVELAND CLIFFS INC 2,149,713 Mining -0.0173

43 FREEPORT MCMORAN COPPER & GOLD 11,781,130 Mining -0.0170

44 ROWAN COMPANIES INC 4,092,873 Mining -0.0170

Table 2. Fifty Firms with the Largest DCoVaR Estimates over 2006-2007

∆CoVaR is estimated as the difference between two linear quantile regressions estimates of the market portfolio return on individual stock returns: the 1 percent quantile estimate less the 50 percent quantile estimate. The reference portfolio is the CRSP equally-weighted market index. Quantile regressions are estimated using the R package Quantreg. The median market capitalization is the median value of the closing share price times the number of shares outstanding.

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4 ACCREDITED HOME LENDERS HLDG CO 697,385 Other Financial -0.0697

5 URANERZ ENERGY CORP 133,479 Mining -0.0688

11 W C I COMMUNITIES INC 792,183 Construction -0.0583

12 COMPUCREDIT CORP 1,786,092 Other Financial -0.0582

13 BEAZER HOMES USA INC 1,601,873 Construction -0.0573

14 EMPIRE RESOURCES INC DEL 104,041 Wholesale Trade -0.0570

15 HOUSTON AMERICAN ENERGY CORP 120,614 Mining -0.0566

16 RADIAN GROUP INC 4,673,175 Insurance -0.0557

17 I C O GLOBAL COMMS HLDGS LTD DE 590,548 Transportation, Communi -0.0555

18 E TRADE FINANCIAL CORP 9,718,507 Broker Dealers -0.0552

19 BUCKEYE TECHNOLOGIES INC 451,391 Manufacturing -0.0548

27 WINN DIXIE STORES INC 1,001,750 Retail Trade -0.0523

28 PANACOS PHARMACEUTICALS INC 241,728 Manufacturing -0.0520

29 CROCS INC 1,864,861 Manufacturing -0.0517

30 GRUBB & ELLIS CO 266,193 Other Financial -0.0514

31 K B W INC 877,814 Broker Dealers -0.0513

32 FRANKLIN BANK CORP 425,906 Depository Institutions -0.0511

33 ASIAINFO HOLDINGS INC 289,401 Services -0.0508

34 FIRST AVENUE NETWORKS INC 613,862 Transportation, Communications -0.0508

35 WHEELING PITTSBURGH CORP 293,215 Manufacturing -0.0505

41 TIENS BIOTECH GROUP USA INC 285,336 Services -0.0493

42 AIRSPAN NETWORKS INC 142,206 Transportation, Communications -0.0492

43 FREEPORT MCMORAN COPPER & GOLD 11,781,130 Mining -0.0490

44 MERITAGE HOMES CORP 1,069,891 Construction -0.0486

45 EDDIE BAUER HOLDINGS INC 273,345 Retail Trade -0.0485

46 A Z Z INC 235,577 Manufacturing -0.0483

47 GRAFTECH INTERNATIONAL LTD 722,492 Manufacturing -0.0483

48 P M I GROUP INC 3,817,586 Insurance -0.0483

49 ASYST TECHNOLOGIES INC 341,391 Manufacturing -0.0480

50 NEUROGEN CORP 213,849 Manufacturing -0.0480

Table 3. Fifty Firms with the Largest MES Estimates over 2006-2007

MES is estimated as the average of individual stock returns on sample subset of days that correspond with the 5 percent worst days of the equally-weighted CRSP stock market index. The median market capitalization is the median value of the closing share price times the number of shares outstanding.

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Table 4: Monte Carlo Small Sample Distribution Estimates for the Critical Values of the DWHT ∆CoVaR and DWHT MES Test

Statistics

Monte Carlo estimates of the hypothesis tests critical values are for a sample size 500 observations. Critical values are estimated as follows. For each replication, the Gauassian parmeters are choosen with equal probability from a set of three representative population values ρjM={.17,.42,.58};

σM={.014,.023,.037}; and σj={.005,.008,.03}. The DHWT ∆CoVaR and DHWT MES hypothesis test statistics are calulated using bootstrap estimates for the nonparametric and parametric estimator variances using 500

bootstrap samples. The critical value estimates are based on 25 thousand Monte Carlo replications.

Table 5: ΔCoVaR Cramer-von Mises Test for Equality of Two Distributions

Against sample Against pooled sample

The underlying systemic risk test statistic is the difference between the nonparameteric and parametric ∆CoVaR estimators scaled by an estimate of the market portfolio return standard deviation in a sample size of 500 observations when the stock and market return portfolio are distriburted bivariate Gaussian. The test statistic sampling distribution is simulated using 5000 replications for each value of the market return standard deviation holding all other distribution parameters constant. The Cramer-von Mises test statistic tests whether two specific samples are drawn from the same underlying distribution (e.g., a market volatility of .10 versus a market volatility of .40) or whether the specific samples are drawn from the same distribution that generated the pooled sample. The reported probability value indicates the probability that the samples are drawn from the same underlying distribution.

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The underlying systemic risk test statistic is the difference between the nonparameteric and parametric MES estimators scaled by an estimate of the individual stock return standard deviation in a sample size of 500 observations when the stock and market return portfolio are distriburted bivariate Gaussian.

The test statistic sampling distribution is simulated using 20000 replications for each value of the market return standard deviation holding all other distribution parameters constant. The Cramer-von Mises test statitic tests whether two specific samples are drawn from the same underlying distribution (e.g., a stock volatility of .10 versus a stock volatility of .40) or whether the specific samples are drawn from the same distribution that generated the pooled sample. The reported probability value indicates the probability that the samples are drawn from the same underlying distribution.

Table 6: MES Cramer-von Mises Test for Equality of Two Distributions

Critical Value of DCoVaR Test Statistic KCoVaR Critical Value of MES Test Statistic, KM ES Table 7: Critical Values for κCoVaR and κMES Test Statistics

In each replication, we draw 500 observations drawn randomly from a bivariate Gaussian distribution with zero mean and the indicated correlation and we estimate the κCoVaR and κM ES test statistics. The 10%, 5%, and 1% critical values estimates are based on 50,000 Monte Carlo replications.

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Broker Dealers 55 3 8 14.5% 22 27 49.1%

Construction 41 1 4 9.8% 4 11 26.8%

Depository Institutions 426 16 40 9.4% 88 158 37.1%

Insurance 138 5 18 13.0% 36 60 43.5%

Manufacturing 1336 75 159 11.9% 132 326 24.4%

Mining 151 22 55 36.4% 11 28 18.5%

Other Financial 74 9 19 25.7% 13 21 28.4%

Public Administration 11 1 2 18.2% 0 6 54.5%

Retail Trade 227 10 27 11.9% 27 63 27.8%

Services 626 38 80 12.8% 50 129 20.6%

Transportation, Communications 320 21 59 18.4% 80 150 46.9%

Wholesale Trade 113 9 25 22.1% 8 24 21.2%

Table 8: Summary of κCoVaR and κMES Hypothesis Test Results

The κCoVaR statistic is statistically significant at 1% (5%) level if the sample estimated value exceeds its critical value for a given correlation level and Type I error of 1% (5%) reported in Table 7. Similarly, the κM ES statistic is statististically significant if the sample estimated value exceeds the critical value reported in Table 7 for a given correlation and Type I error level .

Distribution

Quantiles βj βM Ωjj ΩjM ΩMM αj αM ν ρjM

1% -1.050 0.239 0.400 0.002 0.243 -0.304 -1.611 2.306 0.002

5% -0.408 0.332 0.705 0.079 0.311 -0.165 -1.301 2.805 0.100

25% 0.010 0.398 1.501 0.293 0.387 0.062 -1.056 3.426 0.313

50% 0.197 0.435 2.608 0.445 0.426 0.213 -0.925 3.875 0.454

75% 0.389 0.469 4.234 0.601 0.470 0.382 -0.815 4.480 0.549

95% 0.776 0.522 7.767 0.908 0.552 0.698 -0.629 5.737 0.651

99% 1.189 0.574 11.369 1.137 0.663 0.987 -0.468 7.617 0.713

Table 9: Maximum Likelihood Estimates of Bivariate Skew-t Distribution for 3518 firms

This table summarizes the distribution quantiles of the bivariate skew-t parameter estimates for 3518 daily stock and market return pairs between January 2006 and December 2007. The parameters are estimated for each firm by maximum likelihood using the Adelchi Azzalini's "SN" package in "R". The subscript j (M ) indicates a parameter estimate that is associated with the individual stock return (equally weighted CRSP portfolio return). Parameter ρjM is not estimated but calculated as ρjM= σjMjjσMM)-0.5. Estimates are for 100 times daily log returns.

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𝐿𝑠𝑦𝑚 𝑡 = 2𝑇1(−√(𝜈 + 1)(1 − 𝜌)

√1 + 𝜌 , 𝜐 + 1)

Figure 1: Symmetric T-Distribution Asymptotic Tail Dependence as a Function of Correlation and Degrees of Freedom

The figure plots the asymptotic tail dependence for various levels of degrees of freedom and correlation for the symmetric bivariate t distribution. Asymptotic tail dependence is given by:

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𝐿𝑠𝑘𝑒𝑤−𝑡=

𝑇1(−2𝛼√(𝜈 + 2)(1 + 𝜌)

2 ; 𝜈 + 2) ∗ 2𝑇1(−√(𝜈 + 1)(1 − 𝜌)

√1 + 𝜌 , 𝜐 + 1) 𝑇1(−𝛼(1 + 𝜌)√𝜈 + 1

√1 + 𝛼2(1 − 𝜌2); 𝜈 + 1)

Figure 2: Magnitude of the Asymptotic Left Tail Dependence in a Skew-t Distribution

The figure illustrates the relationship between strength of the asymptotic left tail dependence for the skew-t distribution and the degrees of freedom parameter, 𝜈, correlation parameter, 𝜌 and skewness parameters, 𝛼1= 𝛼2= 𝛼. The distribution’s asymptotic tail dependence, 𝐿𝑠𝑘𝑒𝑤−𝑡 is given by:

where 𝑇1 represents the cumulative distribution function of univariate student t.

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Figure 3: 𝜅𝐶𝑜𝑉𝑎𝑅 Power against Symmetric T-Distribution Alternatives

The figure shows the simulated sampling distributions 𝜅𝐶𝑜𝑉𝑎𝑅 based on a sample size of 500 and 10 thousand Monte Carlo replications, and a Gaussian kernel density estimator. The blue distributions are test statistics calculated from the bivariate Gaussian distribution (𝛼1= 𝛼2= 0, 𝜈 = ∞) using the median parameter values in Table 6 for 𝛽1, 𝛽2, Ω11, Ω22. When 𝜌 varies Ω12

The figure shows the simulated sampling distributions 𝜅𝐶𝑜𝑉𝑎𝑅 based on a sample size of 500 and 10 thousand Monte Carlo replications, and a Gaussian kernel density estimator. The blue distributions are test statistics calculated from the bivariate Gaussian distribution (𝛼1= 𝛼2= 0, 𝜈 = ∞) using the median parameter values in Table 6 for 𝛽1, 𝛽2, Ω11, Ω22. When 𝜌 varies Ω12

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